Time's Barbed Arrow: Irreversibility, Crypticity, and Stored Information

We show why the amount of information communicated between the past and future–the excess entropy–is not in general the amount of information stored in the present–the statistical complexity. This is a puzzle, and a long-standing one, since the latter is what is required for optimal prediction, but the former describes observed behavior. We layout a classification scheme for dynamical systems and stochastic processes that determines when these two quantities are the same or different. We do this by developing closed-form expressions for the excess entropy in terms of optimal causal predictors and retrodictors–the epsilon-machines of computational mechanics. A process’s causal irreversibility and crypticity are key determining properties.